Higher-order quantum Casimir elements generate the center of Drinfeld–Jimbo quantum supergroups

Establish that, for every Drinfeld–Jimbo quantum supergroup U_q(𝔤) (including U_q(gl_{m|n}) and U_q(osp_{m|2n})), the family of higher-order quantum Casimir elements C_{V,ℓ} constructed via the universal R-matrix—indexed by a finite-dimensional representation V and positive integer ℓ—collectively generates the center Z(U_q(𝔤)).

Background

Zhang, Bracken, and Gould introduced higher-order quantum Casimir elements C_{V,ℓ} in the early 1990s using the universal R-matrix. They conjectured that these elements generate the center of Drinfeld–Jimbo quantum (super)groups. This paper confirms corresponding generation results for classical quantum groups of types B, C, and D (extending prior confirmations in type A), but does not address the supergroup case.

Recent developments—such as explicit eigenvalue formulas for higher-order Casimir elements in quantum supergroups (for U_q(gl_{m|n}) and U_q(osp_{m|2n})) and the establishment of a Harish–Chandra isomorphism for quantum superalgebras—provide tools that may facilitate resolving the conjecture for supergroups, which the authors indicate as future work. Thus, verifying that higher-order quantum Casimir elements generate the center in the supergroup setting remains an outstanding conjecture.